jumping champion (idea)

A term coined by John Conway to describe a number which, for some integer n, is the most frequently occurring difference (or "jump") between pairs of consecutive primes less than n. For smallish integers (at least up to 1012), the only jumping champions which appear are 2, 4 and 6. It is conjectured that the only numbers which ever qualify as jumping champions are 2, 4, 6 (=2x3), 30 (=2x3x5) and so on through the list of primorials.